In what follows, the terminology \(\infty \) -category is used as a synonym for quasi-category, one important model of \((\infty ,1)\) -categories, popularized by Lurie [1]}. We denote the large \(\infty \) -category of small \(\infty \) -categories by \({C}\mathrm {at}_\infty \) . The nerve functor \(N\)  from the category \({C}\mathrm {at}\)  of small categories to that of simplicial sets enables us to see \({C}\mathrm {at}\)  as a subcategory of \({C}\mathrm {at}_\infty \) . We routinely suppress the functor \(N\)  to simplify notation. The inclusion of \({C}\mathrm {at}\)  into \({C}\mathrm {at}_\infty \)  admits a left adjoint
\(\mathrm {Ho}: {C}\mathrm {at}_\infty \rightarrow {C}\mathrm {at}\) 
